Broadband subwavelength focusing of light using a passive sink

Broadband subwavelength focusing oflight using a passive sink

Heeso Noh1,2 Sebastien M. Popoff1 and Hui Cao1,∗1 Department of Applied Physics, Yale University, New Haven, CT 06520, USA

2 Department of Nano and Electronic Physics, Kookmin University, Seoul, South Korea∗ [email protected]

Abstract: Optical absorption is usually considered deleterious, somethingto avoid if at all possible. We propose a broadband nanoabsorber thatcompletely eliminates the diffracting wave, resulting in a subwavelengthenhancement of the field. Broadband operation is made possible byengineering the dispersion of the complex dielectric function. The localenhancement can be significantly improved compared to the standard planewave illumination of a metallic nanoparticle. Our numerical simulationshows that an optical pulse as short as 6 fs can be focused to a 11 nmregion. Not only the local field, but also its gradient are greatly enhanced,pointing to applications in ultrafast nonlinear spectroscopy, sensing andcommunication with deep-subwavelength resolution.

© 2013 Optical Society of America

OCIS codes: (030.0030) Coherence and statistical optics; (240.0240) Optics at surfaces.

References and links1. R. Carminati, R. Pierrat, J. de Rosny and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,”

Opt. Lett. 32, 3107–3109 (2002).2. J. de Rosny and M. Fink, “Overcoming the diffraction limit in wave physics using a time-reversal mirror and a

novel acoustic sink,” Phys. Rev. Lett. 89, 124301 (2002).3. P. Kinsler, “Active drains and causality,” Phys. Rev. A 13, 055804 (2011).4. Y. G. Ma, S. Sahebdivan,C. K. Ong, T. Tyc and U. Leonhardt, “Evidence for subwavelength imaging with positive

refraction,” New J. Phys. 13, 033016 (2002).5. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev.

Lett. 105, 053901 (2010).6. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control

of absorption,” Science 331, 889–892 (2011).7. H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorp-

tion,” Phys. Rev. Lett. 108, 186805 (2012).8. G. Montaldo, G. Lerosey, A. Derode, A. Tourin, J. de Rosny, and M. Fink, “Telecommunication in a disordered

environment with iterative time reversal,” Waves in Random Media 14, 287–302 (2004).9. M. Pu, Q. Feng, M. Wang, C. Hu, C. Huang, X. Ma, Z. Zhao, C. Wang, and X. Luo, “Ultrathin broadband nearly

perfect absorber with symmetrical coherent illumination,” Opt. Express 20, 2246–2254 (2012).10. A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Muller, and R. H. Rinkleff, “White-light cavities, atomic

phase coherence, and gravitational wave detectors,” Opt. Commun. 134, 431–439 (1997).11. N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole transi-

tions,” J. Phys. Chem. B 115, 5304–5311 (2011).12. P. A. Belov, C. R. Simovski, and S. A. Tretyakov, “Example of bianisotropic electromagnetic crystals: The spiral

medium,” Phys. Rev. E 67, 056622 (2003).13. A. I. Rahachou and I. V. Zozoulenko, “Light propagation in nanorod arrays,” J. Opt. A: Pure and Applied Optics

9, 265–270 (2007).14. N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, “Ultrasonic metamaterials with

negative modulus,” Nat. Mater. 5, 452–456 (2006).

#187020 - $15.00 USD Received 21 Mar 2013; revised 11 Jun 2013; accepted 27 Jun 2013; published 15 Jul 2013(C) 2013 OSA 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017435 | OPTICS EXPRESS 17435

15. C. Ding, L. Hao, and X. Zhao, “Two-dimensional acoustic metamaterial with negative modulus,” J. Appl. Phys.108, 074911 (2010).

1. Introduction

Overcoming the diffraction limit and focusing waves on a deep-subwavelength scale have beenwidely pursued for light, microwave and acoustics. The tight focusing not only can strongly en-hance linear and nonlinear interactions between waves and matter, but also greatly improve thespatial resolution in sensing, imaging and communication. It is well known that in a homoge-neous medium, wave diffraction limits the size of a focal spot to half the wavelength. This limitcan be understood in a simple case of a monochromatic scalar wave. Consider a spherical wavee−ikr−iωt/r converging to the origin r = 0, where ω is the angular frequency, k = 2π/λ and λis the wavelength. After passing the origin, the wave becomes diverging, of the form eikr−iωt/r.The total field is a superposition of the two, i.e, eikr−iωt/r−e−ikr−iωt/r ∝ sin(kr−ωt)/r, whichis proportional to the imaginary part of Green’s function [1] - the solution to the wave equationin a homogeneous medium. The minus sign of the diverging wave causes a destructive inter-ference of the incoming and outgoing waves, which removes the singularity at r = 0. The focalspot has a width of π/k = λ/2, the well-known diffraction limit. Therefore, diffraction is aconsequence of destructive interference of the converging and diverging waves.

One way of breaking the diffraction limit is to make the interference constructive by flip-ping the sign of outgoing wave, so that the total field becomes eikr−iωt/r + e−ikr−iωt/r ∝cos(kr−ωt)/r. This can be done by placing a resonant scatterer at the origin. Below the reso-nant frequency, the diffracted wave experiences a π phase shift. For the visible light, the surfaceplasmon resonance of a metallic nanosphere might be used. The intrinsic absorption of metal,however, would weaken the interference effect.

An alternative is to utilize the absorption to eliminate the outgoing wave, so the total fieldis equal to the incoming one eikr−iωt/r. This can be realized by placing a perfect absorber atthe origin. The field amplitude, albeit a factor of two smaller than in the first case, approachesinfinity as r → 0. Diffraction is thus overcome, as demonstrated by de Rosny and Fink in anacoustic wave experiment [2]. They used a time-reversed sink to cancel the diffracted wavee−ikr−iωt/r and obtained a focal spot smaller than λ/14. This active sink can operate over abroad frequency range, but it requires prior knowledge of the incoming signal [3] and a setupto generate the time reverse source at the focal spot. The idea of using a sink to remove thediverging wave has recently been extended to microwaves [4].

In this paper, we propose a passive sink which is not driven externally and does not need priorknowledge of the temporal profile of the impinging wave. This method is based on the coherentperfect absorber we recently developed [5–7]. Perfect absorption of coherent light by two-or three-dimensional metallic nanostructures can be achieved via critical coupling to surfaceplasmon resonances [7]. Although it was originally demonstrated for narrow-band operation,we will show here that nearly perfect absorption can be realized over a continuous band offrequency by engineering the dispersion of the complex dielectric function. We demonstrate innumerical simulation that a 6.7 fs pulse is focused from far field to a region of ∼ 13 nm. Thetemporal shape of the pulse can be arbitrary; as long as the spatial wavefront matches the time-reversed radiation pattern of a surface plasmon resonance, the incident pulse will be completelyabsorbed.

The nearly perfect absorption of an ultrashort optical pulse results in strong localization of theelectric field energy in space and time, which greatly enhances light-matter interactions, e.g.,two-photon excitation of molecules. Not only the field amplitude but also the field gradientis enhanced by many orders of magnitude, when the incoming light is critically coupled toa multipole resonance of surface plasmon in the metallic nanoparticle. The steeply varying


field can efficiently excite the dipole-forbidden transitions of molecules and probe the darkstates. Hence, the broad-band nearly perfect absorber may be useful for ultrafast nonlinearspectroscopy and sensing at the nanoscale. Furthermore, our scheme is general and applicableto other types of waves, e.g. acoustic wave, microwave, pointing to possible applications inshock wave lithotripsy, microwave communication with deep-subwavelength resolution [8].The nearly perfect absorption of an ultrashort optical pulse results in strong localization of theelectric field energy in space and time, which greatly enhances light-matter interactions, e.g.,two-photon excitation of molecules. Not only the field amplitude but also the field gradientis enhanced by many orders of magnitude, when the incoming light is critically coupled toa multipole resonance of surface plasmon in the metallic nanoparticle. The steeply varyingfield can efficiently excite the dipole-forbidden transitions of molecules and probe the darkstates. Hence, the broad-band nearly perfect absorber may be useful for ultrafast nonlinearspectroscopy and sensing at the nanoscale. Furthermore, our scheme is general and applicableto other types of waves, e.g. acoustic wave, microwave, pointing to possible applications inshock wave lithotripsy, microwave communication with deep-subwavelength resolution [8].

2. Principle

In principle, if a subwavelength object is placed at r = 0, the outgoing wave can undergo a phaseshift and its amplitude may be reduced. This two effects are respectively due to the internalresonance and dissipation of the object. The destructive interference effect cancels the divergingpart of the field only when the incoming and outgoing waves have the same amplitude and πphase difference. Any deviation from these conditions leads to a subwavelength enhancementof the field. That is why the commonly used plane wave illumination of a metallic nanoparticle,that is dissipative and has resonance close to the optical excitation wavelength, produces astrong local field enhancement. In the introduction section we discussed two extreme scenariosto optimize the subwavelength field enhancement: adjusting only the amplitude or the phase ofthe outgoing wave. In this section, we will consider more general cases.

First, we show how to change the relative phase between the converging and diverging wavesto make them interfere constructively. For simplicity, we limit our analysis to two-dimensional(2D) waves, the extension to 3D is straight-forward [7]. Let us consider a cylindrical wavepropagates in the x− y plane towards the origin; its magnetic field is along the z axis. Afterconverging at the origin, the wave diverges, and the total field can be written in the polar coor-

dinates as Hz(r,θ) = H(2)m (kr)eimθ + sH(1)

m (kr)eimθ , where H(1)m (H(2)

m ) is the mth-order Hankelfunction of the first (second) kind and represents an outgoing (incoming) wave. The complexnumber, s, is the relative amplitude of the outgoing wave; in the absence of dissipation the out-going wave carries the same energy as the incoming wave, |s|= 1. The phase of s is determinedby the solution to the Maxwell’s equations, which gives s = 1 in vacuum or lossless dielec-

tric media. Although both H(1)m (kr) and H(2)

m (kr) diverge at r = 0, the total field remains finite,

because the diverging part of H(1)m (kr) is canceled by that of H(2)

m (kr). This cancellation canbe regarded as a consequence of destructive interference between the incoming and outgoingwaves.

To switch to constructive interference, we must reverse the sign of s. One way isplacing a cylinder of radius R at r = 0 to interact with the incident wave. Due to thecylindrical symmetry of the system, the angular momentum m is conserved upon scat-tering, so is the polarization. Thus the total field outside the cylinder (r > R) has thesame expression as before, but with a different value of s; inside the cylinder (r < R)Hz(r,θ) = aJm(nkr)eimθ , where Jm is the Bessel function of the first kind, n =

√ε , ε

is the complex dielectric function of the cylinder, and a is a normalization constant.By matching the fields at the cylinder surface (r = R), we solve for s and get s =


[nJm(nkR)H(2)m

′(kR)− J′m(nkR)H(2)m (kR)]/[J′m(nkR)H(1)

m (kR)−nJm(nkR)H(1)m

′(kR)], where J′(H ′) is the first-order derivative of the Bessel (Hankel) function. In the following, we will con-sider three different situations: a metallic nanoparticle illuminated by a plane wave (Fig. 1(a)),the same object illuminated by a cylindrical wave that matches a resonant mode of the particle(Fig. 1(b)) and a cylindrical wave in an homogeneous medium (Fig. 1(c)).

(a) (b) (c)

Fig. 1. Different configurations studied. (a) a nanoparticle illuminated by a plane wave. (b)a nanoparticle illuminated by a cylindrical wave. (c) cylindrical wave in an homogeneousmedium.

Setting s = −1 for a particular m, we find the complex ε if k and R are fixed. Alternatively,we can fix ε and solve for k and R. Figure 2 shows the dipole (m = 1) and quadrupole (m = 2)solutions for a cylinder of R = 5.1 nm. The wavelength λ of light in the vacuum outside thecylinder is set at 532 nm. In both cases, the imaginary part of ε (εi) vanishes, the cylinder has noloss and |s| = 1. ε is purely real and negative, representing a lossless metal. In the quasi-staticlimit, surface plasmon resonances occur at ε = −1. Below the resonant frequency (where ε isslightly less than -1), the outgoing wave experiences an additional π phase shift, and s = −1.

The diverging parts of H(1)m (kr) and H(2)

m (kr) no longer cancel each other giving rise to hugelocal field enhancements commonly observed with metallic nanoparticles.

The total field does not diverge due to the finite radius and reaches the maximal value at thesurface of the cylinder (r = R). The smaller the R, the larger the maximal field magnitude. Asshown in Fig. 2, the electric field intensity at r =R is 105 times higher than what can be obtainedby focusing in vacuum without the cylinder for m = 1. Nevertheless, a similar enhancement isobtained by exciting the same particle with a plane wave. When kR � 1, the input energy ispredominantly coupled to the m= 1 mode that has the lowest quality factor and whose radiationpattern has a better overlap with the plane wavefront. The enhancement originates from thesame dipole resonance, thus the behavior is similar. However, for m = 2, the field enhancementat r = R is 1.5× 109, which is about 3 times that with plane wave excitation. The quadrupoleresonance (m = 2) has stronger field enhancement, because of its lower radiative decay rateand higher quality factor. Since the plane wave excitation couples most of the input energyinto the m = 1 resonance that has a lower quality factor than the m = 2 resonance, the fieldenhancement is weaker than that with a cylindrical wave of m = 2. Excitation of higher-order(m > 2) resonances will lead to a further enhancement of local fields.

To describe the size of the focal spot, we use the effective diameter defined as de f f =2∫ ∞

0 (|Er|2 + |Eθ |2)r2dr/∫ ∞

0 (|Er|2 + |Eθ |2rdr. In Fig. 1, de f f = 18 nm for m = 1, and de f f =11 nm for m = 2. These value are approximately λ/40. Without the cylinder, the effective di-ameter of the focal spot is 212 nm, close to λ/2 - the diffraction limit. In fact, the focal spotsize, which is slightly larger than the diameter of the cylinder (10.2 nm), can be further reducedby using a smaller cylinder. However, the value of εr must be set correctly. As shown in Fig.3(a), a slight deviation of εr from the right value makes s a complex number. The phase relationbetween the incoming and outgoing waves is modified, and they no longer interfere construc-tively. The total field can be expressed as the sum of a diverging term and a non-diverging


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Fig. 2. Interference between a focusing wave (λ = 532 nm) and its diffracted wave ismade constructive (s = −1) by placing a lossless metallic nanoparticle (R = 5.1 nm; ε =−1.01 for m = 1; ε = −1.001 for m = 2) at the origin (r = 0). Black solid curves (bluedashed curves) represent the radial distributions of electric field intensity with (without)the metallic scatterer for m = 1 (a,b) and m = 2 (c,d) in the linear scale (a,c) and thelogarithmic scale (b,d). For comparison, the red dotted curves in (a,c) are the intensitieswhen a plane wave is incident onto the metallic scatterer. All the intensities are normalizedto the maximal values that can be obtained by focusing the same incident waves in vacuumwithout the particle. Maximal local field intensities are enhanced 5 orders of magnitude form = 1 and 9 orders of magnitude for m = 2.


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Fig. 3. (a) Relative amplitude (black solid line) and phase (blue dashed line) of thediffracted wave s (λ = 532 nm, m = 1) as a function of the real part of the dielectric con-stant ε of the metallic cylinder (R = 5.1 nm). As soon as the value of Re[ε] deviates fromthat for s=−1, the phase of s changes dramatically, breaking the condition for constructiveinterference of the scattered wave and the incident wave. (b) Maximal electric field inten-sity Ie at the cylinder surface (r = R) as a function of the real part of ε . Ie is normalizedto the maximal field intensity produced by the same incident wave in the absence of thecylinder. When s deviates from -1, Ie drops quickly.

term, Hz(r,θ) = HND(r,θ) + (1− s)HD(r,θ), with HND(r,θ) = H(2)m (kr)eimθ +H(1)

m (kr)eimθ

and HD(r,θ) = H(1)m (kr)eimθ . As s deviates from 1, the maximal field intensity drops quickly

[Fig. 3(b)].To achieve s = −1, the metallic nanocylinder must be lossless (εi = 0), which is difficult to

realize at optical frequency. What happens if εi �= 0? Figure 4(a) shows how the amplitude andphase of s change as we fix εr and gradually increase εi for m = 1. At small εi, the amplitude ofs drops, while the phase stays nearly constant. Hence, the absorption merely reduces the ampli-tude of the outgoing wave, but does not change its phase relation to the incoming one. The twowaves still interfere constructively, albeit not as strongly as in the lossless case and the maxi-mal field intensity decreases [Fig. 4(b)]. When εi reaches a critical value 5.72×10−3, s = 0, theoutgoing wave vanishes, the incoming wave is completely absorbed via critical coupling to thenanocylinder cavity. With a further increase of the εi, the outgoing wave reappears, but with aπ phase shift. The interference between the incoming and outgoing waves becomes destructive,causing a further reduction of the maximal field intensity. The local field enhancement is deter-mined by the diverging term of the total field, whose intensity is proportional to |1− s|2, whichfits well the maximal intensity in Fig. 3(b). Since the destructive interference effect cancelsthe diverging part of the field only when s is very close to 1, the spot size is otherwise mainlygoverned by the huge local field enhancement close to the metallic surface. Nevertheless, asthe imaginary part of ε increases, the relative weight of the diverging term decreases and theeffective diameter of the focal spot gradually increases in Fig. 4(b).

In the above calculation we used the cylindrical wave that perfectly matches the radiationpattern of the resonance of a nanocylinder. If the incident beam is not a full cylindrical wave, itcan be expressed as a linear superposition of multiple cylindrical waves of different azimuthalnumber m. Since only some of these components can meet the condition for perfect absorption,the local field enhancement and sub-diffractcion focusing are still present but weaker than thecases shown above.


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Fig. 4. (a) Relative amplitude (black solid line) and phase (blue dashed line) of thediffracted wave s (λ = 532 nm, m = 1) as a function of the imaginary part of the dielectricconstant ε of the metallic cylinder (R = 5.1 nm). The real part of ε is fixed at -1.01. Asthe amplitude of s reaches zero, its phase experiences a π shift. (b) Maximal electric fieldintensity Ie at the cylinder surface (black solid line) and the focal spot size defined by theeffective diameter de f f (blue dashed line) as a function of the imaginary part of ε . Ie is nor-malized to the maximal field intensity produced by the same incident wave in the absenceof the cylinder. The red dotted curve represents the fit Ie = α |1− s|2, with α = 6.19×104.As the imaginary part of ε increases from 0 to 0.01, Ie decreases monotonically, while theeffective spot size increases .

3. Broadband response

The above study was performed at a single frequency. Is it possible to achieve subwavelengthfocusing for a broad-band pulse? To answer this question, let us consider the case s = 0; the so-lution can be easily extended to other values of s. s = 0 corresponds to the perfect absorption ofthe incoming wave, which has been studied previously for narrow-band operation [7]. It utilizescritical coupling to the surface plasmon resonances in metallic nanostructures to achieve com-plete absorption. Since the surface plasmon resonances exist at discrete frequencies, the perfectabsorption is narrow-band. Recently, an one-dimensional broad-band nearly perfect absorberwas proposed and simulated by utilizing dispersive materials, e.g. the heavily doped silicon [9].

To design a passive sink of subwavelength size for a broad frequency range is not trivialbecause of causality. The causality of a physical system implies that the complex dielectricfunction must be analytic in the upper half of the complex plane. The scattering matrix, thatlinks the incoming eld to the outgoing eld, should also be an analytic function in the upperhalf-plane. A consequence is that the amplitude of the outgoing eld can only be zero at discretefrequencies. Aware of this limitation, we will rst nd the mathematical solution for the complexdielectric function corresponding to s = 0 in a broad frequency range. We then use the Lorentzmodel, that satises the analyticity of the dielectric function, to approximate this solution.

We consider a nanocylinder of R= 5.1 nm. We calculate the complex ε for satisfying s= 0 ateach wavelength λ in the range of 456 nm to 638 nm. Figure 5 plots εr and εi versus k = 2π/λfor m = 1. As k increases, εr decreases, and εi increases. Such dispersion can be realized at thelow frequency side of a material resonance. We fit the dispersive ε by the Lorentz model:

ε( f )(ω) = εinf −ω2

p

ω2 −ω20 + iωΓ0

, (1)

where ω0 is the resonant frequency, Γ0 is the damping constant, ωp is the plasma frequency,


εinf is the non-resonant high-frequency component of the dielectric function.

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Fig. 5. Black solid curves in (a) and (b) are the real and imaginary parts of the dielectricfunction ε of a nanocylinder (R = 5.1 nm) at r = 0 for perfect absorption of a cylindricalwave (m= 1) at every wavelength in the range of 456 nm - 638 nm. The blue dashed curvesrepresents the best fitting of complex dielectric function with the Lorentz model descriedby Eq. (1). The parameters are given in the text.

The dashed lines in Fig. 5(a,b) shows the best fit with ω0 = 5.28× 1015, ωp = 6.67× 1014,Γ0 = 5.85×1015 and εinf =−1.02+0.008i. Such a dispersion relation, the cylinder can nearly

perfectly absorb the incoming waves H(2)m=1(kr) across the wavelength range of 456 nm - 638

nm. Thus it can be considered as a “white-light” nanocavity, in analog to the macroscopicwhite-light cavities utilizing the negative dispersion of atoms [10].

The broadband operation enables a nearly perfect absorption of a short pulse that is spreadover the wavelength range of 456 nm - 638 nm. Figure 6(a) and (b) are movies (online) show-ing the temporal evolution of the field distribution when a Gaussian pulse of width 6.7 fs is

impinging on the cylinder. The magnetic field of the incident pulse is Hz = H(2)m (kr)eimθ f (t),

where f (t) = e(t+r/c)2/3.32e−iωct , t is in the unit of femtosecond, ωc is 2πc/λ (λ = 532nm) and

c is velocity of light. The incident wave has an angular momentum m = 1 (m = 2) in Fig. 6(a)[(b)], exciting the dipole (quadruple) resonance of the surface plasmon in the nanocylinder. Ineither case, the incoming pulse is completely absorbed by the cylinder, resulting in a nanoscalefocal spot. Consequently, a strong local field builds up in the vicinity of the cylinder for a shortperiod of time. Figure 6(b) plots the temporal evolution of the electric field intensity at thecylinder surface Ie(r = R, t). It shows a peak of width 6.7 fs, equal to the incident pulse width.The peak height is 4 (8) orders of magnitude higher than the focused intensity of the same pulsewithout the cylinder for m = 1 (m = 2). As a comparison, we plot the temporal response of anon-dispersive particle that satisfies s = 0 for m = 1 at the central frequency of the pulse. Themaximal amplitude of the pulse is 3.3 times weaker than that with the tailored dispersive parti-cle. The focal spot size is d = 18 nm, 11 nm for m = 1,2. Therefore, the nanocylinder with anappropriate dispersion ε(λ ) enables the deep-subwavelength focusing of a broadband signal,that greatly enhances the local fields.

Note that with the same dispersive ε , optical pulses of arbitrary temporal shape can be com-pletely absorbed, as long as all the spectral components are within the bandwidth of nearlyperfect absorption. We repeat the calculation with chirped pulses, and the incident wave-front remains cylindrical. Figure 6(d) is a movie of a pulse with the chirped temporal form


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Fig. 6. (a,b) Single-frame excerpts from movies (Media 1, Media 2) of the spatial dis-tribution of magnetic field Hz(r,θ) when a Gaussian pulse of width 6.7 fs impinges on ananocylinder (R = 5.1 nm) at the origin (r = 0). The pulse has a cylindrical wavefront andan angular momentum m = 1 in (a), m = 2 in (b). The pulse spectra is centered at λ = 532nm with a FWHM of 180 nm. The dispersive dielectric function of the nanocylinder ischosen to reach a nearly perfect absorption at all incident wavelengths. (c) Electric fieldintensity at the surface of the cylinder (r = R) as a function of time in the case of (a) -black solid line, and (b) - blue dashed line. The red curve represents the field intensitywhen the dielectric function is non-dispersive and perfect absorption is reached only atthe center wavelength λ = 532 nm of the incident pulse for m = 1. The intensity is nor-malized by the maximal intensity of the same incident pulse without the nanocylinder. (d)Single-frame excerpts from movies showing a chirped pulse (Media 3), whose spectrumand spatial wavefront are identical to the Gaussian pulse in (b), is nearly perfectly absorbedby the nanocylinder with the same ε(λ ). The temporal profile of the pulse is described inthe text.


http://www.opticsinfobase.org/oe/viewmedia.cfm?uri=oe-21-15-17421-1



f (t) = e(t+r/c)2/3.32e−i(ωc−0.55t)t , t in the unit of fs, and m = 2 impinging on the nanocylinder.

The pulse is focused to a λ/44 spot via nearly perfect absorption of the diffracted wave.We note that the broadband subwavelength absorber is not a blackbody. The difference is that

the blackbody can perfectly absorb light incident from any direction with arbitrary wavefront,while the near-perfect absorber works only for specific wavefront of the incident light.

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Fig. 7. Two-photon excitation rate Γ2 as a function of the radial coordinate r when a Gaus-sian pulse of width 6.7 fs is nearly perfectly absorbed by a nanocylinder (R = 5.1 nm) atr = 0. The incident pulse has the cylindrical wavefront of m = 1 in (a), m = 2 in (b) (blacksolid line). For comparison, the dashed blue curve represents the incident wave of a planefront. The pulse spectra and the dielectric function of the cylinder in (a) and (b) are identi-cal to those in Fig. 6(a) and (b), respectively. Γ2 is normalized by the maximal two-photonexcitation rate Γ0 of the same input pulse without the nanocylinder.

4. Applications

The subwavelength focusing of a short optical pulse greatly enhance the local field, which maybe utilized to enhance the light matter interaction, especially the nonlinear one. As an example,we consider the two-photon excitation of molecules in the vicinity of a metallic nanocylinder(R = 5.1 nm) under the condition of broadband nearly perfect absorption. The rate of two-photon excitation by a short pulse at a spatial position (r,θ) is Γ2(r,θ) ∝

∫I2e (r,θ , t)dt, where

Ie is the electric field intensity. Due to the cylindrical symmetry of the system and the excitationwave, Γ2 is independent of θ . Figure 7 plots the enhancement of Γ2, i.e. the ratio Γ2(r)/Γ0,where Γ0 is the maximal rate of local two-photon excitation by the same incident pulse inthe absence of the nanocylinder. At the cylinder surface, Γ2/Γ0 = 109 for m = 1, and 3×1016 for m = 2. These enhancements are respectively 30 and 300 times higher than the onesobtained with the same short pulse having a plane wavefront (Γ2/Γ0 = 3×107 and 1014). Suchenhancements occurs only locally (r=R). Moving away from the cylinder surface, Γ2 decreasesrapidly. The spatially-integrated rate, Γ2t = 2π

∫ ∞R Γ2(r)rdr, is proportional to the total intensity

of fluorescence from two-photon excitation. After normalizing it by that without the cylinder,we obtain the total enhancement factor, which is 9.7×107 for m = 1, and 8.1×1015 for m = 2.The enhancement factor can be further increased by using higher-order resonances (m > 2) orsmaller scatterers (R < 5 nm).

In addition to the field amplitude, the field gradient is also greatly enhanced. The steeplyvarying electric field can efficiently excite molecular multipole transitions that are barely ex-


(a) (b)

102

104

106

0 50 100

|∂Er/∂r|

r(nm)

102

104

106

108

0 50 100

|∂Er/∂r|

r(nm)

Fig. 8. Radial gradient of the electric field component in the radial direction Er, in the casea cylindrical wave nearly perfectly absorbed by a nanocylinder (R = 5.1 nm) located atr = 0 (black solid line) and a plane wave excitation of the same cylinder (dashed blue line)at λ = 532 nm. m = 1 in (a), m = 2 in (b). |∂Er/∂ r| is normalized by the maximal gradientwithout the nanocylinder.

cited, if at all, by far-field plane waves. For example, the rate of electric quadrupole transition isproportional to the spatial gradient of the electric field [11]. As an illustration, we calculate theelectric field gradients in the case of perfect absorption at a single wavelength λ = 532 nm. Forthe TM-polarized light, the electric field has the radial component Er(r,θ) and the azimuthalcomponent Eθ (r,θ). There are four gradients, among them the radial gradient of Er has thelargest amplitude. Figure 8 compares |∂Er/∂ r| with a R = 5.1 nm cylinder to that without form = 1 [Fig. 8(a)] and m = 2 [Fig. 8(b)]. At the cylinder surface, |∂Er/∂ r| reaches the maximalvalue, which is seven (six) orders of magnitude higher for m = 2 (m = 1) than the maximalvalue without the cylinder. By comparison, a plane wave excitation at the same wavelengthgives rise to an enhancement of four orders of magnitudes lower in both cases. Such a dramaticenhancement of the field gradient will lead to an efficient excitation of the multiple transitions.

5. Conclusion

In summary, we have proposed and simulated a broadband passive sink that can greatly improvethe subwavelength focusing with a nanoparticle. It relies on the coherent absorption to eliminatethe outgoing (diverging) wave, which would otherwise interfere with the incoming (converging)wave to produce the diffraction pattern that limits the spatial resolution.

The sink is not driven externally, and a broadband operation is made possible by engineer-ing the dispersion of the complex dielectric function. Our numerical simulation shows that anoptical pulse as short as 6 fs can be focused to a 11 nm region. This scheme can be easilyextended to microwave and acoustic wave. The spectral dispersion for broadband response isreachable with carefully-designed metamaterials [12–15]. Such metamaterials based on deepsubwavelength resonators can both have a tailored dispersion relation and a size smaller thanthe wavelength. This media are good candidates for the fabrication of subwavelength nearlyperfect absorbers for microwave and acoustic wave.

The broadband nearly perfect nanoabsorber enables deep-subwavelength focusing of shortoptical pulses, which will greatly enhance nonlinear optical process. The spatial gradient of


the field will also be enhanced dramatically, which may efficiently excite the dark states ofmolecules. Potential applications include ultrafast nonlinear spectroscopy, sensing and com-munication with deep-subwavelength resolution.

Acknowledgments

We thank Mathis Fink, Geoffroy Lerosey and Seng Fatt Liew for useful discussions. This workis funded by the NSF Grants ECCS-1068642.


Broadband subwavelength focusing of light using a passive sink

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